Planck 2015 results: XVIII. Background geometry and topology of the Universe

DOI:10.1051/0004-6361/201525829 c

ESO 2016

Astronomy

&

Astrophysics

Planck 2015 results Special feature

Planck 2015 results

XVIII. Background geometry and topology of the Universe

Planck Collaboration: P. A. R. Ade⁹⁵, N. Aghanim⁶², M. Arnaud⁷⁸, M. Ashdown^74,6, J. Aumont⁶², C. Baccigalupi⁹³, A. J. Banday^105,10, R. B. Barreiro⁶⁹, N. Bartolo^33,70, S. Basak⁹³, E. Battaner^106,107, K. Benabed^63,104, A. Benoît⁶⁰, A. Benoit-Lévy^26,63,104, J.-P. Bernard^105,10, M. Bersanelli^36,50, P. Bielewicz^88,10,93, J. J. Bock^71,12, A. Bonaldi⁷², L. Bonavera²¹, J. R. Bond⁹, J. Borrill^15,98, F. R. Bouchet^63,97, M. Bucher¹,

C. Burigana^49,34,51, R. C. Butler⁴⁹, E. Calabrese¹⁰¹, J.-F. Cardoso^79,1,63, A. Catalano^80,77, A. Challinor^66,74,13, A. Chamballu^78,17,62, H. C. Chiang^29,7, P. R. Christensen^89,39, S. Church¹⁰⁰, D. L. Clements⁵⁸, S. Colombi^63,104, L. P. L. Colombo^25,71, C. Combet⁸⁰, F. Couchot⁷⁶,

A. Coulais⁷⁷, B. P. Crill^71,12, A. Curto^69,6,74, F. Cuttaia⁴⁹, L. Danese⁹³, R. D. Davies⁷², R. J. Davis⁷², P. de Bernardis³⁵, A. de Rosa⁴⁹, G. de Zotti^46,93, J. Delabrouille¹, F.-X. Désert⁵⁶, J. M. Diego⁶⁹, H. Dole^62,61, S. Donzelli⁵⁰, O. Doré^71,12, M. Douspis⁶², A. Ducout^63,58, X. Dupac⁴¹, G. Efstathiou⁶⁶, F. Elsner^26,63,104, T. A. Enßlin⁸⁴, H. K. Eriksen⁶⁷, S. Feeney⁵⁸, J. Fergusson¹³, F. Finelli^49,51, O. Forni^105,10, M. Frailis⁴⁸, A. A. Fraisse²⁹, E. Franceschi⁴⁹, A. Frejsel⁸⁹, S. Galeotta⁴⁸, S. Galli⁷³, K. Ganga¹, M. Giard^105,10, Y. Giraud-Héraud¹, E. Gjerløw⁶⁷,

J. González-Nuevo^21,69, K. M. Górski^71,108, S. Gratton^74,66, A. Gregorio^37,48,55, A. Gruppuso^49,51, J. E. Gudmundsson^102,91,29, F. K. Hansen⁶⁷, D. Hanson^85,71,9, D. L. Harrison^66,74, S. Henrot-Versillé⁷⁶, C. Hernández-Monteagudo^14,84, D. Herranz⁶⁹, S. R. Hildebrandt^71,12, E. Hivon^63,104,

M. Hobson⁶, W. A. Holmes⁷¹, A. Hornstrup¹⁸, W. Hovest⁸⁴, K. M. Huffenberger²⁷, G. Hurier⁶², A. H. Jaffe^58,?, T. R. Jaffe^105,10, W. C. Jones²⁹, M. Juvela²⁸, E. Keihänen²⁸, R. Keskitalo¹⁵, T. S. Kisner⁸², J. Knoche⁸⁴, M. Kunz^19,62,3, H. Kurki-Suonio^28,45, G. Lagache^5,62, A. Lähteenmäki^2,45,

J.-M. Lamarre⁷⁷, A. Lasenby^6,74, M. Lattanzi^34,52, C. R. Lawrence⁷¹, R. Leonardi⁸, J. Lesgourgues^64,103, F. Levrier⁷⁷, M. Liguori^33,70, P. B. Lilje⁶⁷, M. Linden-Vørnle¹⁸, M. López-Caniego⁴¹, P. M. Lubin³¹, J. F. Macías-Pérez⁸⁰, G. Maggio⁴⁸, D. Maino^36,50, N. Mandolesi^49,34,

A. Mangilli^62,76, M. Maris⁴⁸, P. G. Martin⁹, E. Martínez-González⁶⁹, S. Masi³⁵, S. Matarrese^33,70,43, J. D. McEwen⁸⁶, P. McGehee⁵⁹, P. R. Meinhold³¹, A. Melchiorri^35,53, L. Mendes⁴¹, A. Mennella^36,50, M. Migliaccio^66,74, S. Mitra^57,71, M.-A. Miville-Deschênes^62,9, A. Moneti⁶³,

L. Montier^105,10, G. Morgante⁴⁹, D. Mortlock⁵⁸, A. Moss⁹⁶, D. Munshi⁹⁵, J. A. Murphy⁸⁷, P. Naselsky^90,40, F. Nati²⁹, P. Natoli^34,4,52, C. B. Netterfield²², H. U. Nørgaard-Nielsen¹⁸, F. Noviello⁷², D. Novikov⁸³, I. Novikov^89,83, C. A. Oxborrow¹⁸, F. Paci⁹³, L. Pagano^35,53, F. Pajot⁶², D. Paoletti^49,51, F. Pasian⁴⁸, G. Patanchon¹, H. V. Peiris²⁶, O. Perdereau⁷⁶, L. Perotto⁸⁰, F. Perrotta⁹³, V. Pettorino⁴⁴, F. Piacentini³⁵,

M. Piat¹, E. Pierpaoli²⁵, D. Pietrobon⁷¹, S. Plaszczynski⁷⁶, D. Pogosyan³⁰, E. Pointecouteau^105,10, G. Polenta^4,47, L. Popa⁶⁵, G. W. Pratt⁷⁸, G. Prézeau^12,71, S. Prunet^63,104, J.-L. Puget⁶², J. P. Rachen^23,84, R. Rebolo^68,16,20, M. Reinecke⁸⁴, M. Remazeilles^72,62,1, C. Renault⁸⁰, A. Renzi^38,54, I. Ristorcelli^105,10, G. Rocha^71,12, C. Rosset¹, M. Rossetti^36,50, G. Roudier^1,77,71, M. Rowan-Robinson⁵⁸, J. A. Rubiño-Martín^68,20,

B. Rusholme⁵⁹, M. Sandri⁴⁹, D. Santos⁸⁰, M. Savelainen^28,45, G. Savini⁹², D. Scott²⁴, M. D. Seiffert^71,12, E. P. S. Shellard¹³, L. D. Spencer⁹⁵, V. Stolyarov^6,99,75, R. Stompor¹, R. Sudiwala⁹⁵, D. Sutton^66,74, A.-S. Suur-Uski^28,45, J.-F. Sygnet⁶³, J. A. Tauber⁴², L. Terenzi^94,49, L. Toffolatti^21,69,49, M. Tomasi^36,50, M. Tristram⁷⁶, M. Tucci¹⁹, J. Tuovinen¹¹, L. Valenziano⁴⁹, J. Valiviita^28,45, F. Van Tent⁸¹, P. Vielva⁶⁹,

F. Villa⁴⁹, L. A. Wade⁷¹, B. D. Wandelt^63,104,32, I. K. Wehus^71,67, D. Yvon¹⁷, A. Zacchei⁴⁸, and A. Zonca³¹ (Affiliations can be found after the references)

Received 6 February 2015/Accepted 10 April 2016 ABSTRACT

Maps of cosmic microwave background (CMB) temperature and polarization from the 2015 release ofPlanckdata provide the highest quality full-sky view of the surface of last scattering available to date. This enables us to detect possible departures from a globally isotropic cosmology.

We present the first searches using CMB polarization for correlations induced by a possible non-trivial topology with a fundamental domain that intersects, or nearly intersects, the last-scattering surface (at comoving distanceχrec), both via a direct scan for matched circular patterns at the intersections and by an optimal likelihood calculation for specific topologies. We specialize to flat spaces with cubic toroidal (T3) and slab (T1) topologies, finding that explicit searches for the latter are sensitive to other topologies with antipodal symmetry. These searches yield no detection of a compact topology with a scale below the diameter of the last-scattering surface. The limits on the radiusR_iof the largest sphere inscribed in the fundamental domain (at log-likelihood ratio∆lnL>−5 relative to a simply-connected flatPlanckbest-fit model) are:R_i >0.97χrecfor the T3 cubic torus; andR_i>0.56χrecfor the T1 slab. The limit for the T3 cubic torus from the matched-circles search is numerically equivalent, R_i >0.97χrecat 99% confidence level from polarization data alone. We also perform a Bayesian search for an anisotropic global Bianchi VIIh

geometry. In the non-physical setting, where the Bianchi cosmology is decoupled from the standard cosmology,Plancktemperature data favour the inclusion of a Bianchi component with a Bayes factor of at least 2.3 units of log-evidence. However, the cosmological parameters that generate this pattern are in strong disagreement with those found from CMB anisotropy data alone. Fitting the induced polarization pattern for this model to thePlanckdata requires an amplitude of−0.10±0.04 compared to the value of+1 if the model were to be correct. In the physically motivated setting, where the Bianchi parameters are coupled and fitted simultaneously with the standard cosmological parameters, we find no evidence for a Bianchi VIIhcosmology and constrain the vorticity of such models to (ω/H)0<7.6×10⁻¹⁰(95% CL).

Key words. cosmic background radiation – cosmology: observations – cosmological parameters – gravitation – methods: data analysis – methods: statistical

? Corresponding author: A. H. Jaffe,[email protected]

1. Introduction

This paper, one of a series associated with the 2015 release of Planck¹ data, will present limits on departures from the global isotropy of spacetime. We assess anisotropic but homogeneous Bianchi cosmological models and non-trivial global topologies in the light of the latest temperature and polarization data.

InPlanck Collaboration XXVI(2014), the limits came from the 2013Planck cosmological data release: cosmic microwave background (CMB) intensity data collected over approximately one year. This work uses the 2015Planckdata: CMB intensity from the whole mission along with a subset of polarization data.

The greater volume of intensity data will allow more restrictive limits on the possibility of topological scales that are slightly larger than the volume enclosed by the last-scattering surface (roughly the Hubble volume), probing the excess anisotropic correlations that would be induced at large angular scales were such a model to obtain. For cubic torus topologies, we can there- fore observe explicit repeated patterns (matched circles) when the comoving length of an edge is less than twice the distance to the recombination surface, χ_rec ' 3.1H₀⁻¹ (using units with c = 1 here and throughout). Polarization, on the other hand, which is largely generated during recombination itself, can pro- vide a more sensitive probe of topological domains smaller than the Hubble volume.

Whereas the analysis of temperature data in multiply con- nected universes has been treated in some depth in the literature (seePlanck Collaboration XXVI 2014, and references therein), the discussion of polarization has been less complete. This paper therefore extends our previous likelihood analysis to polarized data, updates the direct search for matched circles (Cornish et al.

2004) as discussed inBielewicz et al.(2012), and uses these to present the first limits on global topology from polarized CMB data.

The cosmological properties of Bianchi models (Collins & Hawking 1973; Barrow et al. 1985), were ini- tially discussed in the context of CMB intensity (Barrow 1986; Jaffe et al. 2006c,a; Pontzen 2009). As discussed in Planck Collaboration XXVI (2014), it is by now well known that the observed large-scale intensity pattern mimics that of a particular Bianchi VIIhmodel, albeit one with cosmological pa- rameters that are quite different from those needed to reproduce other CMB and cosmological data. More recently the induced polarization patterns have been calculated (Pontzen & Challinor 2007;Pontzen 2009;Pontzen & Challinor 2011). In this paper, we analyse the completePlanckintensity data, and compare the polarization pattern induced by that anisotropic model toPlanck polarization data.

We note that the lack of a strong detection of cosmic B-mode polarization already provides some information about the Bianchi models: the induced geometrical focusing does not distinguish betweenEandBand thus should produce compara- ble amounts of each (e.g., Pontzen 2009). This does not apply to topological models: the linear evolution of primordial pertur- bations guarantees that a lack of primordial tensor perturbations results in a lack ofB-mode polarization – the transfer function is not altered by topology.

1 Planck(http://www.esa.int/Planck) is a project of the Euro- pean Space Agency (ESA) with instruments provided by two scientific consortia funded by ESA member states and led by Principal Investi- gators from France and Italy, telescope reflectors provided through a collaboration between ESA and a scientific consortium led and funded by Denmark, and additional contributions from NASA (USA).

In Sect. 2, we discuss previous limits on anisotropic mod- els from Planck and other experiments. In Sect. 3 we discuss the CMB signals generated in such models, generalized to both temperature and polarization. In Sect.4we describe thePlanck data and simulations we use in this study, the different methods we apply to those data, and the validation checks performed on those simulations. In Sect.5we discuss the results and conclude in Sect.6with the outlook for application of these techniques to future data and broader classes of models.

2. Previous results

The first searches for non-trivial topology on cosmic scales looked for repeated patterns or individual objects in the distri- bution of galaxies (Sokolov & Shvartsman 1974; Fang & Sato 1983; Fagundes & Wichoski 1987; Lehoucq et al. 1996;

Roukema 1996; Weatherley et al. 2003; Fujii & Yoshii 2011).

Searches for topology using the CMB began with COBE (Bennett et al. 1996) and found no indications of a non- trivial topology on the scale of the last-scattering surface (e.g., Starobinskij 1993; Sokolov 1993; Stevens et al. 1993;

de Oliveira-Costa & Smoot 1995;Levin et al. 1998;Bond et al.

1998b,2000;Rocha et al. 2004; but see alsoRoukema 2000b,a).

With the higher resolution and sensitivity of WMAP, there were indications of low power on large scales which could have had a topological origin (Jarosik et al. 2011; Luminet et al. 2003;

Caillerie et al. 2007; Aurich 1999; Aurich et al. 2004, 2005, 2006, 2008; Aurich & Lustig 2013; Lew & Roukema 2008;

Roukema et al. 2008;Niarchou et al. 2004), but this possibility was not borne out by detailed real- and harmonic-space anal- yses in two dimensions (Cornish et al. 2004; Key et al. 2007;

Bielewicz & Riazuelo 2009; Dineen et al. 2005; Kunz et al.

2006; Phillips & Kogut 2006; Niarchou & Jaffe 2007). Most studies, including this work, have emphasized searches for fundamental domains with antipodal correlations; see Vaudrevange et al.(2012) for results from a general search for the patterns induced by non-trivial topology on scales within the volume defined by the last-scattering surface, and, for example, Aurich & Lustig(2014) for a recent discussion of other possible topologies.

For a more complete overview of the field, we direct the reader toPlanck Collaboration XXVI(2014). In that work, we applied various techniques to the Planck 2013 intensity data.

For topology, we showed that a fundamental topological domain smaller than the Hubble volume is strongly disfavoured. This was done in two ways: first, a direct likelihood calculation of specific topological models; and second, a search for the ex- pected repeated “circles in the sky” (Cornish et al. 2004), cali- brated by simply-connected simulations. Both of these showed that the scale of any possible topology must exceed roughly the distance to the last-scattering surface, χrec. For the cubic torus, we found that the radius of the largest sphere inscribed in the topological fundamental domain must beR_i > 0.92χrec(at log-likelihood ratio∆lnL > −5 relative to a simply-connected flat Planck 2013 best-fit model). The matched-circle limit on topologies predicting back-to-back circles wasR_i >0.94χ_recat the 99% confidence level.

Prior to the present work, there have been some extensions of the search for cosmic topology to polarization data. In par- ticular,Bielewicz et al.(2012; see alsoRiazuelo et al. 2006) ex- tended the direct search for matched circles to polarized data and found that the available WMAP data had insufficient sensitivity to provide useful constraints.

For Bianchi VIIh models, in Planck Collaboration XXVI (2014) a full Bayesian analysis of thePlanck2013 temperature data was performed, following the methods of McEwen et al.

(2013). It was concluded that a physically-motivated model was not favoured by the data. If considered as a phenomenologi- cal template (for which the parameters common to the standard stochastic CMB and the deterministic Bianchi VII_h component are not linked), it was shown that an unphysical Bianchi VIIh

model is favoured, with a log-Bayes factor between 1.5±0.1 and 2.8±0.1 – equivalent to an odds ratio of between approximately 1:4 and 1:16 – depending of the component separation technique adopted. Prior to the analysis of Planck Collaboration XXVI (2014), numerous analyses of Bianchi models using COBE (Bennett et al. 1996) and WMAP (Jarosik et al. 2011) data had been performed (Bunn et al. 1996;Kogut et al. 1997;Jaffe et al.

2005, 2006a,c,b; Cayón et al. 2006; Land & Magueijo 2006;

McEwen et al. 2006; Bridges et al. 2007; Ghosh et al. 2007;

Pontzen & Challinor 2007; Bridges et al. 2008; McEwen et al.

2013), and a similar Bianchi template was found in the WMAP data, first by Jaffe et al. (2005) and then subsequently by others (Bridges et al. 2007; Bridges et al. 2008; McEwen et al.

2013). Pontzen & Challinor (2007) discussed the CMB polar- ization signal from Bianchi models, and showed some in- compatibility with WMAP data due to the large amplitude of both E- and B-mode components. For a more detailed re- view of the analysis of Bianchi models we refer the reader to Planck Collaboration XXVI(2014).

3. CMB signals in anisotropic

and multiply-connected universes 3.1. Topology

There is a long history of studying the possible topo- logical compactification of Friedmann-Lemaître-Robertson- Walker (FLRW) cosmologies; we refer readers to overviews such as Levin (2002), Lachieze-Rey & Luminet (1995), and Riazuelo et al. (2004a,b) for mathematical and physical detail.

The effect of a non-trivial topology is equivalent to considering the full (simply-connected) three dimensional spatial slice of the manifold (the covering space) as being tiled by identical repe- titions of a shape which is finite in one or more directions, the fundamental domain. In flat universes, to which we specialize here, there are a finite number of possibilities, each described by one or more continuous parameters describing the size in differ- ent directions.

In this paper, we pay special attention to topological models in which the fundamental domain is a right-rectangular prism (the three-torus, also referred to as “T3”), possibly with one or two infinite dimensions (the T2 “chimney” or “rod”, and T1

“slab” models). We limit these models in a number of ways. We explicitly compute the likelihood of the length of the fundamen- tal domain for the cubic torus. Furthermore, we consider the slab model as a proxy for other models in which the matched circles (or excess correlations) are antipodally aligned, similar to the

“lens” spaces available in manifolds with constant positive cur- vature. These models are thus sensitive to tori with varying side lengths, including those with non-right-angle corners. In these cases, the likelihood would have multiple peaks, one for each of the aligned pairs; their sizes correspond to those of the funda- mental domains and their relative orientation to the angles. These non-rectangular prisms will be discussed in more detail in Jaffe

& Starkman (in prep.).

3.1.1. Computing the covariance matrices

In Planck Collaboration XXVI (2014) we computed the temperature-temperature (T T) covariance matrices by summing up all modes k_nthat are present given the boundary conditions imposed by the non-trivial topology. For a cubic torus, we have a three-dimensional wave vector k_n = (2π/L)n for a triplet of integersn, with unit vector kˆ and the harmonic-space covariance matrix

C^mm_``0^{0(T T)}∝X

n

∆^(T)_{`</sub> (kn,∆η)∆<sup>(T)</sup><sub>`}⁰ (kn,∆η)P(kn)Y`m(kˆ)Y<sub>`</sub>^∗0m⁰(kˆ), (1) where∆^(T)_` (k,∆η) is the temperature radiation transfer function (see, e.g., Bond & Efstathiou 1987; and Seljak & Zaldarriaga 1996).

It is straightforward to extend this method to include polar- ization, since the cubic topology affects neither the local physics that governs the transfer functions, nor the photon propagation.

The only effect is the discretization of the modes. We can there- fore simply replace the radiation transfer function for the tem- perature fluctuations with the one for polarization, and obtain C^mm_``0^0(XX0)∝X

n

∆^(X)_{`</sub> (k<sub>n</sub>,∆η)∆<sup>(X</sup><sub>`}0⁰⁾(k_n,∆η)P(k_n)Y_{`m</sub>(kˆ)Y<sub>`}^∗0m⁰(kˆ), (2) where X,X⁰ = E,T. We are justified in ignoring the possi- bility of B-mode polarization as it is sourced only by primor- dial gravitational radiation even in the presence of non-trivial topology. In this way we obtain three sets of covariance ma- trices:T T, T E, and EE. In addition, since the publication of Planck Collaboration XXVI(2014) we have optimized the cubic torus calculation by taking into account more of the symmetries.

The resulting speed-up of about an order of magnitude allowed us to reach a higher resolution of`_max=64.

The fiducial cosmology assumed in the calculation of the co- variance matrices is a flatΛCDM FLRW Universe with Hubble constantH0 = 100hkm s⁻¹Mpc⁻¹, where:h = 0.6719; scalar spectral index ns = 0.9635; baryon density Ωbh² = 0.0221;

cold dark matter densityΩch² = 0.1197; and neutrino density Ωνh²=0.0006.

3.1.2. Relative information in the matrices

To assess the information content of the covariance matrices, we consider the Kullback-Leibler (KL) divergence (see, e.g., Kunz et al. 2006,2008; and Planck Collaboration XXVI 2014, for further applications of the KL divergence to topology). The KL divergence between two probability distributions p1(x) and p2(x) is given by

dKL=Z

p1(x) lnp1(x)

p2(x) dx. (3)

If the two distributions are Gaussian with covariance matrices C1andC2, this expression simplifies to

dKL=−1 2

hln C1C⁻¹2

+Tr

I−C1C⁻¹2

i, (4)

and is thus an asymmetric measure of the discrepancy between the covariance matrices. The KL divergence can be interpreted as the ensemble average of the log-likelihood ratio∆lnLbetween realizations of the two distributions. Hence, it enables us to probe

the ability to tell if, on average, we can distinguish realizations of p1 from a fixed p2 without having to perform a brute-force Monte Carlo integration. Thus, the KL divergence is related to ensemble averages of the likelihood-ratio plots that we present for simulations (Sect.4.4.1) and real data (Sect.5.1) but can be calculated from the covariance matrices alone. Note that with this definition, the KL divergence is minimized for cases with the best match (maximallikelihood).

InPlanck Collaboration XXVI(2014) we used the KL diver- gence to show that the likelihood is robust to differences in the cosmological model and small differences in the topology.

In Fig. 1 we plot the KL divergence relative to an infi- nite Universe for the slab topology as a function of resolu- tion `max (upper panel) and fundamental domain size (lower panel). Our ability to detect a topology with a fundamental do- main smaller than the distance to the last-scattering surface (ap- proximately at the horizon distance χ<sub>rec</sub> = 3.1H<sub>0</sub><sup>−1</sup>, so with sides of length L = 2χ<sub>rec</sub> = 6.2H<sub>0</sub><sup>−1</sup>) grows significantly with the resolution even beyond the cases that we studied. For the noise levels of the 2015 lowP data considered here and defined in Planck Collaboration XIII(2016), polarization maps do not add much information beyond that contained in the temperature maps, although, as also shown in Sect.4.4.1, the higher sensitiv- ity achievable by the fullPlancklow-`data over all frequencies should enable even stronger constraints on these small funda- mental domains.

If, however, the fundamental domain is larger than the hori- zon (as is the case forL=6.5H⁻¹₀ ) then the relative information in the covariance matrix saturates quite early and a resolution of

`_max'48 is actually sufficient. The main goal is thus to ensure that we have enough discriminatory power right up to the hori- zon size. In addition, polarization does not add much information in this case, irrespective of the noise level. This is to be expected:

polarization is only generated for a short period of time around the surface of last scattering. Once the fundamental domain ex- ceeds the horizon size, the relative information drops rapidly to- wards zero, and the dependence on`_maxbecomes weak.

In Fig.2we plot the KL divergence as a function of the size of the fundamental domain for fixed cube (T3), rod (T2), and slab (T1) topologies, each with fundamental domain size L = 5.5H₀⁻¹, compared to the slab. Each shows a strong dip at L = 5.5H₀⁻¹, indicating the ability to detect this topology (although note the presence of a weaker dip around half the correct size, L'2.75H₀⁻¹). The figure also shows that`max=40 still shows the dip at the correct location, although somewhat more weakly than`max=80.

Note that the shape of the curves is essentially identical, with the slab likelihood able to detect one or more sets of antipodal matched circles (and their related excess correlations at large an- gular scales) present in each case. Figure2therefore shows that using the covariance matrix for a slab (T1) topology also allows detection of rod (T2) and cubic (T3) topologies: this is advan- tageous as the slab covariance matrix is considerably easier to calculate than the cube and rod, since it is only discretized in a single direction. Figure3shows the KL divergence as a function of the relative rotation of the fundamental domain, showing that, despite the lack of the full set of three pairs of antipodal cor- relations, we can determine the relative rotation of a single pair.

This is exactly how the matched-circles tests work. Furthermore, as we will demonstrate in Sect.4.4.1, slab likelihoods are indeed separately sensitive to the different sets of antipodal circles in cubic spaces. We can hence adopt the slab as the most general tool for searching for spaces with antipodal circles.

20 40 60 80

`_max 100101dKLvsinfiniteuniverse

T1 slab

T,L= 6/H0

T+PL= 6/H0

T / T+P,L= 6.5/H0

0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 (H₀L)⁻¹

100101102dKLvsinfiniteuniverse

T1 slab T data only

`max= 48

`max= 56

`max= 64

`max= 72

`max= 80

`max= 88

Fig. 1.KL divergence of slab (T1) topologies relative to an infinite Uni- verse as a function of`maxwith sizesL =6H<sup>−1</sup><sub>0</sub> andL=6.5H<sup>−1</sup><sub>0</sub> (top), and as a function of sizeLof the fundamental domain for various`max

(bottom). A torus withL>6.2H₀⁻¹, corresponding to (H0L)⁻¹<0.154, has a fundamental domain that is larger than the distance to the last- scattering surface and leaves only a small trace in the CMB. This is why the KL divergence drops rapidly at this point. Note that the infor- mation forL=6H₀⁻¹continues to rise with`maxwhereas it levels offfor the slightly largerL=6.5H₀⁻¹case. In thelower panelwe see that there is a slight feature indKLat about half the horizon distance, which is probably due to harmonic effects. The corresponding figures for cubic (T3) topologies look qualitatively similar except that alldKLvalues are three times larger.

3.2. Bianchi models

The polarization properties of Bianchi models were first derived in Pontzen & Challinor (2007) and extensively categorized in Pontzen(2009) andPontzen & Challinor(2011). In these works it was shown that advection in Bianchi universes leads to effi- cient conversion ofE-mode polarization to Bmodes; evidence for a significant Bianchi component found in temperature data would therefore suggest a largeB-mode signal (but not neces- sarily require it; see Pontzen 2009). For examples of the tem- perature and polarization signatures of Bianchi VII_hmodels we refer the reader to Fig. 1 ofPontzen(2009). Despite the poten- tial for CMB polarization to constrain the Bianchi sector, a full

0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 (H0L)⁻¹

020406080100120140160dKL,T1vsothermodels

T1 (slab) T data only

T1,L= 5.5/H0,`max= 40 T1,L= 5.5/H0,`max= 80 T3,L= 5.5/H0,`max= 40 T3,L= 5.5/H0,`max= 80 T2 (2013),L= 5.5/H0,`max= 40

Fig. 2.KL divergence of fixed cubic, rod, and slab topologies with fun- damental domain sideL=5.5H₀⁻¹compared to a slab of variable fun- damental domain sizeL. The chimney space T2 dates from the 2013 analysis (Planck Collaboration XXVI 2014) and was computed for the best-fit parameters of that release. In all cases the smallest KL diver- gence, corresponding to the best fit, appears atL=5.5H₀⁻¹, indicating that the slab space can be used to detect other topologies. An additional dip atL'5.5/(2H0) may be due to a harmonic effect at half the size of the fundamental domain; it is, however, much smaller than the drop in KL divergence at the size of the fundamental domain.

0 1 2 3 4 5 6

ϕ 4550556065707580dKL,T1vsT3

Fig. 3.KL divergence of a slab space relative to a cubic topology, as a function of rotation angle of the slab space (blue curve). Both spaces haveL=5.5H⁻₀¹and`max=80. The horizontal black dashed line gives the KL divergence of an infinite Universe relative to the cubic topol- ogy and illustrates how much better the slab space fits with the correct orientation relative to the cubic torus.

polarization analysis has not yet been carried out. The analy- sis ofPontzen & Challinor (2007) remains the state-of-the-art, where WMAPBBandEBpower spectra were used to demon- strate (using a simple χ² analysis) that a Bianchi VII_h model derived from temperature data was disfavoured compared to an isotropic model.

The subdominant, deterministic CMB contributions of Bianchi VIIhmodels can be characterized by seven parameters:

the matter and dark energy densities,ΩmandΩ_Λ, respectively;

the present dimensionless vorticity, (ω/H)0; the dimension- less length-scale parameter, x, which controls the “tightness”

of the characteristic Bianchi spirals; and the Euler angles², (α, β, γ), describing their orientation (i.e., the choice of coor- dinate system), where H is the Hubble parameter. For further details see Planck Collaboration XXVI (2014), McEwen et al.

(2013),Pontzen(2009),Pontzen & Challinor(2007),Jaffe et al.

(2006c),Jaffe et al.(2005), andBarrow et al.(1985).

4. Methods 4.1. Data

In this work we use data from the Planck 2015 release. This includes intensity maps from the full mission, along with a sub- set of polarization data. Specifically, for the likelihood calcula- tions discussed below (Sect. 4.3.1for application to topology and Sect. 4.3.3 for Bianchi models) which rely on HEALPix maps at Nside = 16, we use the data designated “lowT,P”, as defined for the low-` Planck likelihood for isotropic mod- els (Planck Collaboration XI 2016; Planck Collaboration XIII 2016): lowP polarization maps based on the LFI 70 GHz chan- nel and lowT temperature maps created by the Commander component separation method, along with the appropriate mask and noise covariance matrix. As inPlanck Collaboration XXVI (2014), the intensity noise contribution is negligible on these scales, and diagonal regularizing noise with varianceσ<sup>2</sup><sub>I</sub> =4µK<sup>2</sup> has therefore been added to the intensity portion of the noise covariance matrix. We cut contaminated regions of the sky us- ing the low-`mask defined for thePlanck isotropic likelihood code (Planck Collaboration XI 2016), retaining 94% of the sky for temperature, and the lowT,P polarization mask, cleaned with the templates created from Planck 30 GHz and 353 GHz data, retaining 47% of the sky for polarization.

The matched-circle search (Sects. 4.2 and 5.1.1) uses four component-separated maps (Planck Collaboration IX 2016) which effectively combine both intensity and polarization in- formation from different scales. The maps are smoothed with a Gaussian filter of 30⁰ and 50⁰ full width at half maximum (FWHM) for temperature and polarization, respectively, and de- graded toNside=512. Corresponding temperature and polariza- tion common masks for diffuse emission, with a point source cut for the brightest sources, downgraded analogously to the maps, are used. After degradation, and accounting for the needed ex- pansion of the polarization mask due to the conversion of Q andU toE, the temperature map retains 74% of the sky and the polarization map 40%. These E-mode maps are calculated using the method ofBielewicz et al.(2012; see alsoKim 2011) and correspond to the spherical Laplacian of the scalarE, con- sequently filtering out power at large angular scales.

4.2. Topology: matched circles

As inPlanck Collaboration XXVI(2014), we use the circle com- parison statistic of Cornish et al. (1998), optimized for small- scale anisotropies (Cornish et al. 2004), to search for correlated circles in sky maps of the CMB temperature and polarization anisotropy. The circle comparison statistic uses the fact that the intersection of the topological fundamental domain with the sur- face of last scattering is a circle, potentially viewed from dif- ferent directions in a multiply-connected Universe. Contrary to

2 The activezyzEuler convention is adopted, corresponding to the ro- tation of a physical body in afixedcoordinate system about thez,y, and zaxes byγ,β, andα, respectively.

the temperature anisotropy, sourced by multiple terms at the last-scattering surface (i.e., the internal photon density fluctua- tions combined with the ordinary Sachs-Wolfe and Doppler ef- fects), the CMB polarization anisotropy is sourced only by the quadrupole distribution of radiation scattering from free elec- trons at the moment of recombination (e.g., Kosowsky 1996).

In particular, the recombination signal from polarization is only generated for a short time while there are enough electrons to scatter the photons but few enough for the plasma to be suffi- ciently transparent. Thus, in a multi-connected Universe the po- larization signal does not exhibit the same cancellation of con- tributions from different terms as in the temperature anisotropy (Bielewicz et al. 2012). Polarization thus can provide a better op- portunity for the detection of topological signatures than a tem- perature anisotropy map. There is a small subtlety here: whereas the intensity is a scalar and thus is unchanged when viewed from different directions, the polarization is a tensor which behaves differently under rotation. The polarization pattern itself depends on the viewing angle; hence, we need to use the coordinate- independent quantities,E andB, which are scalars (or pseudo- scalars) and are thus unchanged when viewed from different directions.

The decomposition into E and B of an arbitrary masked CMB polarization map, contaminated by noise, foregrounds, and systematic errors, is itself a computationally demanding task, non-local on the sky. Assuming negligible initial B po- larization, we use only the Emaps produced from component- separated CMB polarization maps using the same approach as Bielewicz et al.(2012).

Compared with the likelihood method described below, the circles search uses higher-resolution maps, and thus is sen- sitive out to a much higher maximum multipole, `<sub>max</sub>. It is also potentially less sensitive to large-scale systematic errors, as the lowest multipoles are effectively filtered out: the po- larization signal is weighted by a factor proportional to `² in the transformation from the Stokes parameters QandU to an E-mode map. From the results of Sect.3.1.2, this indicates that it uses more of the information available when confronting mod- els with fundamental domains within the last-scattering surface compared to our implementation of the likelihood, limited to

`max ' 40. As we show in Sect.4.4.1, this also allows the use of high-pass filtered component-separated maps (as defined in Planck Collaboration IX 2016) without a significant decrease in the ability to detect a multiply-connected topology.

The matched-circle statistic is defined by S⁺_i,j(α, φ∗)=2P

m|m|X_i,mX^∗_j,me^−imφ∗ P

n|n|

|Xi,n|²+|Xj,n|² , (5) whereX_i,mandX_j,mdenote the Fourier coefficients of the temper- ature orE-mode fluctuations around two circles of angular radius αcentred at different points on the sky,iandj, respectively, with relative phase φ_∗. The mth harmonic of the field anisotropies around the circle is weighted by the factor |m|, taking into ac- count the number of degrees of freedom per mode. Such weight- ing enhances the contribution of small-scale structure relative to large-scale fluctuations.

TheS⁺statistic corresponds to pairs of circles with the points ordered in a clockwise direction (phased). For the alternative ordering, when the points are ordered in an anti-clockwise di- rection (anti-phased along one of the circles), the Fourier coef- ficients X_i,m are complex conjugated, defining theS⁻ statistic.

This allows the detection of both orientable and non-orientable topologies. For orientable topologies the matched circles have

anti-phased correlations, while for non-orientable topologies they have a mixture of anti-phased and phased correlations.

TheS^±statistics take values over the interval [−1,1]. Circles that are perfectly matched haveS =1, while uncorrelated circles will have a mean value ofS = 0. To find matched circles for each radiusα, the maximum valueS^±_max(α)=max_i,j,φ∗S^±_i,j(α, φ_∗) is determined.

Because general searches for matched circles are compu- tationally very intensive, we restrict our analysis to a search for pairs of circles centred around antipodal points, so called back-to-back circles. The maps are also downgraded as de- scribed in Sect.4.1. This increases the signal-to-noise ratio and greatly speeds up the computations required, but with no signif- icant loss of discriminatory power. Regions most contaminated by Galactic foreground were removed from the analysis using the common temperature or polarization mask. More details on the numerical implementation of the algorithm can be found in Bielewicz & Banday(2011) andBielewicz et al.(2012).

To draw any conclusions from an analysis based on the statis- ticS^±_max(α), it is very important to correctly estimate the thresh- old for a statistically significant match of circle pairs. We used 300 Monte Carlo simulations of thePlanck SMICAmaps pro- cessed in the same way as the data to establish the threshold such that fewer than 1% of simulations would yield a false event. Note that we perform the entire analysis, including the final statistical calibration, separately for temperature and polarization.

4.3. Likelihood 4.3.1. Topology

For the likelihood analysis of the large-angle intensity and po- larization data we have generalized the method implemented in Planck Collaboration XXVI(2014) to include polarization. The likelihood, i.e., the probability to find a combined temperature and polarization data mapdwith associated noise matrixNgiven a certain topological modelT is then given by

P(d|C[ΘC,ΘT,T],A, ϕ)

∝ 1

√|AC+N|exp

"

−1

2d^∗(AC+N)⁻¹d

#

, (6)

where now d is a 3Npix-component data vector obtained by concatenation of the (I,Q,U) data sets while C and N are 3Npix×3Npix theoretical signal and noise covariance matrices, arranged in block form as

C=











CII CIQ CIU

CQI CQQ CQU

CU I CU Q CUU











, N=











NII NIQ NIU

NQI NQQ NQU

NU I NU Q NUU









 . (7)

Finally,ΘCis the set of standard cosmological parameters,ΘTis the set of topological parameters (e.g., the size,L, of the funda- mental domain),ϕis the orientation of the topology (e.g., the Euler angles), and A is a single amplitude, scaling the signal covariance matrix (this is equivalent to an overall amplitude in front of the power spectrum in the isotropic case). Working in pixel space allows for the straightforward application of an arbi- trary mask, including separate masks for intensity and polariza- tion parts of the data. The masking procedure can also be used to limit the analysis to intensity or polarization only.

SinceC+Nin pixel space is generally poorly conditioned, we again (following the 2013 procedure) project the data vector and covariance matrices onto a limited set of orthonormal basis

vectors, selectNmsuch modes for comparison, and consider the likelihood marginalized over the remainder of the modes, p(d|C[ΘC,ΘT,T], ϕ,A)∝

√ 1

|AC+N|_Mexp











−1 2

N_m

X

n=1

d^∗_n(AC+N)⁻¹nn⁰dn⁰











, (8)

whereCandNare restricted to theN_m×N_msubspace.

The choice of the basis modes and their numberNmused for analysis is a compromise between robust invertibility ofC+N and the amount of information retained. All the models for which likelihoods are compared must be expanded in the same set of modes. Thus, inPlanck Collaboration XXVI(2014) we used the set of eigenmodes of the cut-sky covariance matrix of the fiducial best-fit simply-connected Universe, Cfid, as the analysis basis, limiting ourselves to theNmmodes with the largest eigenvalues.

For comparison with the numbers we use, a full-sky tempera- ture map with maximum multipole`maxhas (`max+1)²−4 inde- pendent modes (four are removed to account for the unobserved monopole and dipole).

The addition of polarization data, with much lower signal- to-noise than the temperature, raises a new question: how is the temperature and polarization data mix reflected in the limited basis set we project onto? The most natural choice is the set of eigenmodes of the signal-to-noise matrix CfidN⁻¹ for the fidu- cial model, and a restriction of the mode set based on signal- to-noise eigenvalues (see, e.g., Bond et al. 1998a). This, how- ever, requires robust invertibility of the noise covariance matrix, which, again, is generally not the case for the smoothed data.

Moreover, such a ranking by S/N would inevitably favour the temperature data, and we wish to explore the effect of includ- ing polarization data on an equal footing with temperature. We therefore continue to use the eigenmodes of the cut-sky fiducial covariance matrix as our basis. By default, we select the firstNm

=1085 eigenmodes (corresponding to`max = 32), though we vary the mode count where it is informative to do so.

In Fig.4we showI,Q, andUmaps of the highest-eigenvalue (i.e., highest contribution to the signal covariance) mode for our fiducial simply-connected model. Note that the scale is different for temperature compared to the two polarization maps: the tem- perature contribution to the mode is much greater than that of either polarization component. We show modes for the masked sky, although in fact the structure at large scales is similar to the full-sky case, rotated and adjusted somewhat to account for the mask. In Fig.5we show the structure of mode 301, with much lower signal amplitude (this particular mode was selected at ran- dom to indicate the relative ratios of temperature, polarization, and noise). Temperature remains dominant, although polariza- tion begins to have a greater effect. Note that at this level of sig- nal amplitude, the pattern is aligned with the mask, and shows a strong correlation between temperature and polarization.

4.3.2. Evaluating the topological likelihood

The aim of the topological likelihood analysis is to calculate the likelihood as a function of the parameters pertaining to a particu- lar topology,p(d|ΘT,T). To do so, we must marginalize over the other parameters appearing in Eq. (8), namelyΘC,ϕ, andA, as p(d|ΘT,T)=

Z

dΘCdϕdA p(d|C[ΘC,ΘT,T], ϕ,A)p(ΘC, ϕ,A). (9) The complexity of the topological covariance matrix calcu- lation precludes a joint examination of the full cosmological

−0.03536 0.03536

−0.000076 0.000076

−0.000076 0.000076

Fig. 4.Mode structure plotted as maps for the eigenvector correspond- ing to the highest-signal eigenvalue of the fiducial simply-connected model. Thetopmap corresponds to temperature,middletoQpolariza- tion, andbottomtoUpolarization. Masked pixels are plotted in grey.

and topological parameter spaces. Instead, we adopt the delta- function prior p(ΘC)=δ(ΘC−Θ^?_C) to fix the cosmological pa- rameters at their fiducial values,Θ^?_C (as defined in Sect.3.1.1), and evaluate the likelihood on a grid of topological parameters using a restricted set of pre-calculated covariance matrices. We note that, as discussed inPlanck Collaboration XXVI(2014), the ability to detect or rule out a multiply connected topology is in- sensitive to the values of the cosmological parameters adopted for the calculation of the covariance matrices.

−0.0826 0.0826

−0.0001184 0.0001184

−0.0001184 0.0001184

Fig. 5. Mode structure plotted as maps for the eigenvector corre- sponding to the 301st-highest-signal eigenvalue of the fiducial simply- connected model. Thetopmap corresponds to temperature,middletoQ polarization, andbottomtoUpolarization. Masked pixels are plotted in grey.

In the setting described above, Eq. (9) simplifies to p(d|ΘT,i,T)=Z

dϕdA p(d|C[Θ^?C,ΘT,i,T], ϕ,A)p(ϕ,A), (10) where the likelihood at each gridpoint in topological parameter space,ΘT,i, is equal to the probability of obtaining the data given fixed cosmological and topological parameters and a compact- ification (i.e., fundamental domain shape and size), marginal- ized over orientation and amplitude. The calculation therefore reduces to evaluating the Bayesian evidence for a set of gridded topologies. As we focus on cubic torus and slab topologies in

this work, we note that the sole topological parameter of interest is the size of the fundamental domain,L.

Even after fixing the cosmological parameters, calculat- ing the Bayesian evidence is a time-consuming process, and is further complicated by the multimodal likelihood functions typical in non-trivial topologies. We therefore approach the problem on two fronts. We first approximate the likelihood function using a “profile likelihood” approach, as presented in Planck Collaboration XXVI (2014), in which the marginaliza- tion in Eq. (10) is replaced with maximization in the four- dimensional space of orientation and amplitude parameters.

Specifically, we maximize the likelihood over the three angles defining the orientation of the fundamental domain using a three- dimensional Amoeba search (e.g., Press et al. 1992), where at each orientation the likelihood is separately maximized over the amplitude. Due to the complex structure of the likelihood surface in orientation space, we repeat this procedure five times with different starting orientations. This number of repetitions was chosen as a compromise between computational efficiency and assurance of statistical robustness, after testing of various strate- gies for the number of repetitions and the distribution of starting points, along with explicit extra runs to test outliers. To ensure uniform and non-degenerate coverage, the orientation space is traversed in a Cartesian projection of the northern hemisphere of the three-sphereS3 representation of rotations.

The profile likelihood calculation allows rapid evaluation of the likelihood and testing of different models compared with a variety of data and simulations, but it is difficult to interpret in a Bayesian setting. As we show below, however, the numerical results of profiling over this limited set of parameters agree nu- merically very well with the statistically correct marginalization procedure.

Our second approach explicitly calculates the marginalized likelihood, Eq. (10), allowing full Bayesian inference at the cost of increased computation time. We use the publicMultiNest³ code (Feroz & Hobson 2008; Feroz et al. 2009, 2013) −opti- mized for exploring multimodal probability distributions in tens of dimensions − to compute the desired evidence values via nested sampling (Skilling 2004).MultiNestis run in its impor- tance nested sampling mode (Feroz et al. 2013) using 200 live points, with tolerance and efficiency set to their recommended values of 0.5 and 0.3, respectively. The final ingredient needed to calculate the evidence values are priors for the marginalized parameters. We use a log prior for the amplitude, truncated to the range 0.1 ≤A ≤10, and the Euler angles are defined to be uniform in 0≤α <2π,−1≤cosβ≤1, and 0≤γ <2π, respec- tively;MultiNestis able to wrap the priors onαandγ. The combined code will be made public as part of theAniCosmo⁴ package (McEwen et al. 2013).

It is worth noting that this formalism can be extended to compare models with different compactifications (or the simply connected model) using Bayesian model selection: the only ad- ditional requirements are priors for the topological parameters.

Taking the current slab and cubic torus topologies as examples, by defining a prior on the size of the fundamental domain one can calculate the evidence for each model. Assuming each topol- ogy is equally likely a priori, i.e., thatp(Tslab)=p(Tcub), one can then write down the relative probability of the two topologies given the data:

p(T_slab|d) p(Tcub|d) '

P

ip(L_i|T_slab)p(d|L_i,T_slab) P

jp(L⁰_j|Tcub)p(d|L⁰_j,Tcub)· (11)

3 http://www.mrao.cam.ac.uk/software/multinest/

4 http://www.jasonmcewen.org/

Unfortunately, it is difficult to provide a physically-motivated proper prior distribution for the size of the fundamental domain.

Even pleading ignorance and choosing a “naïve” uniform prior would require an arbitrary upper limit to L whose exact value would strongly influence the final conclusion. For this reason, we refrain from extending the formalism to model selection within this manuscript.

4.3.3. Bianchi models

While physically the cosmological densities describing Bianchi models should be identified with their standardΛCDM counter- parts, in previous analyses unphysical models have been con- sidered in which the densities are allowed to differ. The first coherent analysis of Bianchi VIIh models was performed by McEwen et al.(2013), where theΛCDM and Bianchi densities are coupled and all cosmological and Bianchi parameters are fit simultaneously. In the analysis of Planck Collaboration XXVI (2014), in order to compare with all prior studies both coupled and decoupled models were analysed. We consider the same two models here: namely, the physicalopen-coupled-Bianchimodel where an open cosmology is considered (for consistency with the open Bianchi VIIh models), in which the Bianchi densi- ties are coupled to their standard cosmological counterparts; and the phenomenologicalflat-decoupled-Bianchimodel where a flat cosmology is considered and in which the Bianchi densities are decoupled.

We firstly carry out a full Bayesian analysis for these two Bianchi VIIh models, repeating the analysis per- formed in Planck Collaboration XXVI (2014) with updated Planck temperature data. The methodology is described in detail in McEwen et al. (2013) and summarized in Planck Collaboration XXVI (2014). The complete poste- rior distribution of all Bianchi and cosmological parameters is sampled and Bayesian evidence values are computed to compare Bianchi VIIhmodels to their concordance counterparts. Bianchi temperature signatures are simulated using theBianchi2⁵code (McEwen et al. 2013), while the AniCosmo code is used to perform the analysis, which in turn usesMultiNestto sample the posterior distribution and compute evidence values.

To connect with polarization data, we secondly analyse polarization templates computed using the best-fit parameters from the analysis of temperature data. For the resulting small set of best-fit models, polarization templates are computed us- ing the approach of Pontzen & Challinor (2007) and Pontzen (2009), and have been provided by Pontzen (priv. comm.).

These Bianchi VIIh simulations are more accurate than those considered for the temperature analyses performed here and in previous works (see, e.g.,Planck Collaboration XXVI 2014;

McEwen et al. 2013; Bridges et al. 2008; Bridges et al. 2007;

Jaffe et al. 2005,2006a,b,c), since the recombination history is modelled. The overall morphology of the patterns are consistent between the codes; the strongest effect of incorporating the re- combination history is its impact on the polarization fraction, although the amplitude of the temperature component can also vary by approximately 5% (which is calibrated in the current analysis, as described below).

Using the simulated Bianchi VII_h polarization templates computed following Pontzen & Challinor (2007) and Pontzen (2009), and provided by Pontzen (priv. comm.), we per- form a maximum-likelihood fit for the amplitude of these templates using Planck polarization data (a full Bayesian

5 http://www.jasonmcewen.org/

evidence calculation of the complete temperature and polariza- tion data set incorporating the more accurate Bianchi models of Pontzen & Challinor 2007; and Pontzen 2009, is left to fu- ture work). The likelihood in the Bianchi scenario is identi- cal to that considered inPlanck Collaboration XXVI(2014) and McEwen et al.(2013); however, we now consider the Bianchi and cosmological parameters fixed and simply introduce a scal- ing of the Bianchi template. The resulting likelihood reads:

P(d|λ,t)∝exp

−1

2(d−λt)^†(C+N)⁻¹(d−λt)

, (12)

whereddenotes the data vector,t =b(Θ^?_B) is the Bianchi tem- plate for best-fit Bianchi parameters Θ^?_B, C = C(ΘC) is the cosmological covariance matrix for the best-fit cosmological pa- rametersΘ^?_C,Nis the noise covariance, andλis the introduced scaling parameter (the effective vorticity of the scaled Bianchi component is simplyλ(ω/H)₀).

In order to effectively handle noise and partial sky coverage the data are analysed in pixel space. We restrict to polarization data only here since temperature data are used to determine the best-fit Bianchi parameters. The data and template vectors thus contain unmaskedQandUStokes components only and, corre- spondingly, the cosmological and noise covariance matrices are given by the polarization (QandU) subspace of Eq. (7), and again contain unmasked pixels only.

The maximum-likelihood (ML) estimate of the template am- plitude is given byλ^ML = t^†(C+N)⁻¹d/h

t^†(C+N)⁻¹ti and its dispersion by∆λ^ML=[t^†(C+N)⁻¹t]^−1/2(see, e.g.,Kogut et al.

1997;Jaffe et al. 2005). If Planckpolarization data support the best-fit Bianchi model found from the analysis of temperature data we would expectλ^ML '1. A statistically significant devi- ation from unity in the fitted amplitude can thus be used to rule out the Bianchi model using polarization data.

As highlighted above, different methods are used to simu- late Bianchi temperature and polarization components, where the amplitude of the temperature component may vary by a few percent between methods. We calibrate out this amplitude mis- match by scaling the polarization components by a multiplica- tive factor fitted so that the temperature components simulated by the two methods match, using a maximum-likelihood tem- plate fit again, as described above.

4.4. Simulations and validation 4.4.1. Topology

Matched circles. Before beginning the search for pairs of matched circles in the Planck data, we validate our al- gorithm using the same simulations as employed for the Planck Collaboration XXVI(2014) andBielewicz et al. (2012) papers, i.e., the CMB sky for a Universe with a three-torus topol- ogy for which the dimension of the cubic fundamental domain isL = 2H₀⁻¹, well within the last-scattering surface. We com- puted thea`mcoefficients up to the multipole of order` =500 and convolved them with the same smoothing beam profile as used for thePlanck SMICA map. To the map was added noise corresponding to theSMICAmap. In particular, we verified that our code is able to find all pairs of matched circles in such a map. The statistic S_max⁻ (α) for the E-mode map is shown in Fig.6.

Because for the baseline analysis we use high-pass filtered maps, we also show the analysis of theSMICAE-mode map high- pass filtered so that the lowest order multipoles (` < 20) are